Influid dynamics, thedrag equation is a formula used to calculate the force ofdrag experienced by an object due to movement through a fully enclosingfluid. The equation is:$${\displaystyle F_{\mathrm{d}}=\frac{1}{2}\rho u^2{c}_{\mathrm{d}}A}$$where

• ${\displaystyle F_{\mathrm{d}}}$ is the dragforce, which is by definition the force component in the direction of the flow velocity,
• ${\displaystyle \rho }$ is themass density of the fluid,^[1]
• ${\displaystyle u}$ is theflow velocity relative to the object,
• ${\displaystyle A}$ is the referencearea, and
• ${\displaystyle c_{\mathrm{d}}}$ is thedrag coefficient – adimensionlesscoefficient related to the object's geometry.

The equation is attributed toLord Rayleigh, who originally usedL² in place ofA (withL being some linear dimension).^[2]

The reference areaA is typically defined as the area of theorthographic projection of the object on a plane perpendicular to the direction of motion. For non-hollow objects with simple shape, such as a sphere, this is exactly the same as the maximalcross sectional area. For other objects (for instance, a rolling tube or the body of a cyclist),A may be significantly larger than the area of any cross section along any plane perpendicular to the direction of motion.Airfoils use the square of thechord length as the reference area; since airfoil chords are usually defined with a length of 1, the reference area is also 1. Aircraft use the wing area (or rotor-blade area) as the reference area, which makes for an easy comparison tolift.Airships andbodies of revolution use the volumetric coefficient of drag, in which the reference area is the square of thecube root of the airship's volume. Sometimes different reference areas are given for the same object in which case a drag coefficient corresponding to each of these different areas must be given.

The drag coefficient${\displaystyle c_{\mathrm{d}}}$ is defined in combination with the choice of reference area and captures bothskin friction andform drag. If the fluid is a liquid,${\displaystyle c_{\mathrm{d}}}$ depends on theReynolds number; if the fluid is a gas,${\displaystyle c_{\mathrm{d}}}$ depends on both the Reynolds number and theMach number.

For sharp-corneredbluff bodies, like square cylinders and plates held transverse to the flow direction, this equation is applicable with the drag coefficient as a constant value when theReynolds number is greater than 1000.^[3] For smooth bodies, like a cylinder, the drag coefficient may vary significantly until Reynolds numbers up to 10⁷ (ten million).^[4]

The equation is easier understood for the idealized situation where all of the fluid impinges on the reference area and comes to a complete stop, building upstagnation pressure over the whole area. No real object exactly corresponds to this behavior.${\displaystyle c_{\mathrm{d}}}$ is the ratio of drag for any real object to that of the ideal object. In practice a rough un-streamlined body (a bluff body) will have a${\displaystyle c_{\mathrm{d}}}$ around 1, more or less. Smoother objects can have much lower values of${\displaystyle c_{\mathrm{d}}}$. The equation is precise – it simply provides the definition of${\displaystyle c_{\mathrm{d}}}$ (drag coefficient), which varies with theReynolds number and is found by experiment.

Of particular importance is the${\displaystyle u^2}$ dependence on flow velocity, meaning that fluid drag increases with the square of flow velocity. When flow velocity is doubled, for example, not only does the fluid strike with twice the flow velocity, but twice themass of fluid strikes per second. Therefore, the change ofmomentum per time, i.e. theforce experienced, is multiplied by four. This is in contrast with solid-on-soliddynamic friction, which generally has very little velocity dependence.

The drag force can also be specified as$${\displaystyle F_{\mathrm{d}}\propto P_{\mathrm{D}}A}$$whereP_D is the pressure exerted by the fluid on areaA. Here the pressureP_D is referred to asdynamic pressure due to thekinetic energy of the fluid experiencing relative flow velocityu. This is defined in similar form as the kinetic energy equation:$${\displaystyle P_{\mathrm{D}}=\frac{1}{2}\rho u^2}$$

Thedrag equation may be derived to within a multiplicative constant by the method ofdimensional analysis. If a moving fluid meets an object, it exerts a force on the object. Suppose that the fluid is a liquid, and the variables involved – under some conditions – are the:

• speedu,
• fluid densityρ,
• kinematic viscosityν of the fluid,
• size of the body, expressed in terms of its wetted areaA, and
• drag forceF_d.

Using the algorithm of theBuckingham π theorem, these five variables can be reduced to two dimensionless groups:

• drag coefficient c_d and
• Reynolds number Re.

That this is so becomes apparent when the drag forceF_d is expressed as part of a function of the other variables in the problem:

$${\displaystyle f_a(F_{\mathrm{d}},u,A,\rho ,\nu )=0.}$$

This rather odd form of expression is used because it does not assume a one-to-one relationship. Here,f_a is some (as-yet-unknown) function that takes five arguments. Now the right-hand side is zero in any system of units; so it should be possible to express the relationship described byf_a in terms of only dimensionless groups.

There are many ways of combining the five arguments off_a to form dimensionless groups, but the Buckingham π theorem states that there will be two such groups. The most appropriate are the Reynolds number, given by

$${\displaystyle \mathrm{R}\mathrm{e}=\frac{u\sqrt{A}}{\nu }}$$

and the drag coefficient, given by

$${\displaystyle c_{\mathrm{d}}=\frac{F_{\mathrm{d}}}{\frac{1}{2}\rho A{u}^2}.}$$

Thus the function of five variables may be replaced by another function of only two variables:

$${\displaystyle f_b\left(\frac{F_{\mathrm{d}}}{\frac{1}{2}\rho A{u}^2},\frac{u\sqrt{A}}{\nu }\right)=0.}$$

wheref_b is some function of two arguments.The original law is then reduced to a law involving only these two numbers.

Because the only unknown in the above equation is the drag forceF_d, it is possible to express it as

Thus the force is simply⁠1/2⁠ρAu² times some (as-yet-unknown) functionf_c of the Reynolds number Re – a considerably simpler system than the original five-argument function given above.

Dimensional analysis thus makes a very complex problem (trying to determine the behavior of a function of five variables) a much simpler one: the determination of the drag as a function of only one variable, the Reynolds number.

If the fluid is a gas, certain properties of the gas influence the drag and those properties must also be taken into account. Those properties are conventionally considered to be the absolute temperature of the gas, and the ratio of its specific heats. These two properties determine the speed of sound in the gas at its given temperature. The Buckingham pi theorem then leads to a third dimensionless group, the ratio of therelative velocity to the speed of sound, which is known as theMach number. Consequently when a body is moving relative to a gas, the drag coefficient varies with the Mach number and the Reynolds number.

The analysis also gives other information for free, so to speak. The analysis shows that, other things being equal, the drag force will be proportional to the density of the fluid. This kind of information often proves to be extremely valuable, especially in the early stages of a research project.

To empirically determine the Reynolds number dependence, instead of experimenting on a large body with fast-flowing fluids (such as real-size airplanes inwind tunnels), one may just as well experiment using a small model in a flow of higher velocity because these two systems deliversimilitude by having the same Reynolds number. If the same Reynolds number and Mach number cannot be achieved just by using a flow of higher velocity it may be advantageous to use a fluid of greater density or lower viscosity.^[5]

• Aerodynamic drag
• Angle of attack
• Morison equation
• Newton's sine-square law of air resistance
• Stall (flight)
• Terminal velocity

• Batchelor, G.K. (1967).An Introduction to Fluid Dynamics. Cambridge University Press.ISBN 0-521-66396-2.
• Huntley, H. E. (1967).Dimensional Analysis. Dover. LOC 67-17978.
• Benson, Tom."Falling Object with Air Resistance". US: NASA. RetrievedJune 9, 2022.
• Benson, Tom."The drag equation". US: NASA. RetrievedJune 9, 2022.

• Drag (physics)
• Equations of fluid dynamics
• Aircraft wing design

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